A Preliminaries 1
1 Introduction for teachers 3
² Purpose and intended audience, 3 ² Topics in the
book, 6 ² Why pluralism?, 13 ² Feedback, 18 ² Acknowledgments, 19
2 Introduction for students 20
² Who should study logic?, 20 ² Formalism and certi¯-
cation, 25 ² Language and levels, 34 ² Semantics and
syntactics, 39 ² Historical perspective, 49 ² Pluralism, 57
² Jarden's example (optional), 63
3 Informal set theory 65
² Sets and their members, 68 ² Russell's paradox, 77 ² Subsets, 79 ² Functions, 84 ² The Axiom of Choice
(optional), 92 ² Operations on sets, 94 ² Venn dia-
grams, 102 ² Syllogisms (optional), 111 ² In¯nite sets
(postponable), 116
4 Topologies and interiors (postponable) 126
² Topologies, 127 ² Interiors, 133 ² Generated topologies
and ¯nite topologies (optional), 139
5 English and informal classical logic 146
² Language and bias, 146 ² Parts of speech, 150 ² Semantic values, 151 ² Disjunction (or), 152 ² Con-
junction (and), 155 ² Negation (not), 156 ² Material
implication, 161 ² Cotenability, fusion, and constants
vi Contents
(postponable), 170 ² Methods of proof, 174 ² Working
backwards, 177 ² Quanti¯ers, 183 ² Induction, 195 ² Induction examples (optional), 199
6 De¯nition of a formal language 206
² The alphabet, 206 ² The grammar, 210 ² Removing
parentheses, 215 ² De¯ned symbols, 219 ² Pre¯x
notation (optional), 220 ² Variable sharing, 221 ² Formula schemes, 222 ² Order preserving or reversing
subformulas (postponable), 228
B Semantics 233
7 De¯nitions for semantics 235
² Interpretations, 235 ² Functional interpretations, 237
² Tautology and truth preservation, 240
8 Numerically valued interpretations 245
² The two-valued interpretation, 245 ² Fuzzy interpre-
tations, 251 ² Two integer-valued interpretations, 258
² More about comparative logic, 262 ² More about
Sugihara's interpretation, 263
9 Set-valued interpretations 269
² Powerset interpretations, 269 ² Hexagon interpretation
(optional), 272 ² The crystal interpretation, 273 ² Church's diamond (optional), 277
10 Topological semantics (postponable) 281
² Topological interpretations, 281 ² Examples, 282 ² Common tautologies, 285 ² Nonredundancy of sym-
bols, 286 ² Variable sharing, 289 ² Adequacy of ¯nite
topologies (optional), 290 ² Disjunction property (op-
tional), 293
Contents vii
11 More advanced topics in semantics 295
² Common tautologies, 295 ² Images of interpreta-
tions, 301 ² Dugundji formulas, 307
C Basic syntactics 311
12 Inference systems 313
13 Basic implication 318
² Assumptions of basic implication, 319 ² A few easy
derivations, 320 ² Lemmaless expansions, 326 ² De-
tachmental corollaries, 330 ² Iterated implication (post-
ponable), 332
14 Basic logic 336
² Further assumptions, 336 ² Basic positive logic, 339
² Basic negation, 341 ² Substitution principles, 343
D One-formula extensions 349
15 Contraction 351
² Weak contraction, 351 ² Contraction, 355
16 Expansion and positive paradox 357
² Expansion and mingle, 357 ² Positive paradox (strong
expansion), 359 ² Further consequences of positive para-
dox, 362
17 Explosion 365
18 Fusion 369
19 Not-elimination 372
² Not-elimination and contrapositives, 372 ² Interchange-
ability results, 373 ² Miscellaneous consequences of not-
elimination, 375
viii Contents
20 Relativity 377
E Soundness and major logics 381
21 Soundness 383
22 Constructive axioms: avoiding not-elimination 385
² Constructive implication, 386 ² Herbrand-Tarski De-
duction Principle, 387 ² Basic logic revisited, 393 ² Soundness, 397 ² Nonconstructive axioms and classical
logic, 399 ² Glivenko's Principle, 402
23 Relevant axioms: avoiding expansion 405
² Some syntactic results, 405 ² Relevant deduction
principle (optional), 407 ² Soundness, 408 ² Mingle:
slightly irrelevant, 411 ² Positive paradox and classical
logic, 415
24 Fuzzy axioms: avoiding contraction 417
² Axioms, 417 ² Meredith's chain proof, 419 ² Addi-
tional notations, 421 ² Wajsberg logic, 422 ² Deduction
principle for Wajsberg logic, 426
25 Classical logic 430
² Axioms, 430 ² Soundness results, 431 ² Independence
of axioms, 431
26 Abelian logic 437
F Advanced results 441
27 Harrop's principle for constructive logic 443
² Meyer's valuation, 443 ² Harrop's principle, 448 ² The disjunction property, 451 ² Admissibility, 451 ² Results in other logics, 452
Contents ix
28 Multiple worlds for implications 454
² Multiple worlds, 454 ² Implication models, 458 ² Soundness, 460 ² Canonical models, 461 ² Complete-
ness, 464
29 Completeness via maximality 466
² Maximal unproving sets, 466 ² Classical logic, 470
² Wajsberg logic, 477 ² Constructive logic, 479 ² Non-¯nitely-axiomatizable logics, 485
References 487
Symbol list 493
Index 495
A Preliminaries 1
1 Introduction for teachers 3
² Purpose and intended audience, 3 ² Topics in the
book, 6 ² Why pluralism?, 13 ² Feedback, 18 ² Acknowledgments, 19
2 Introduction for students 20
² Who should study logic?, 20 ² Formalism and certi¯-
cation, 25 ² Language and levels, 34 ² Semantics and
syntactics, 39 ² Historical perspective, 49 ² Pluralism, 57
² Jarden's example (optional), 63
3 Informal set theory 65
² Sets and their members, 68 ² Russell's paradox, 77 ² Subsets, 79 ² Functions, 84 ² The Axiom of Choice
(optional), 92 ² Operations on sets, 94 ² Venn dia-
grams, 102 ² Syllogisms (optional), 111 ² In¯nite sets
(postponable), 116
4 Topologies and interiors (postponable) 126
² Topologies, 127 ² Interiors, 133 ² Generated topologies
and ¯nite topologies (optional), 139
5 English and informal classical logic 146
² Language and bias, 146 ² Parts of speech, 150 ² Semantic values, 151 ² Disjunction (or), 152 ² Con-
junction (and), 155 ² Negation (not), 156 ² Material
implication, 161 ² Cotenability, fusion, and constants
vi Contents
(postponable), 170 ² Methods of proof, 174 ² Working
backwards, 177 ² Quanti¯ers, 183 ² Induction, 195 ² Induction examples (optional), 199
6 De¯nition of a formal language 206
² The alphabet, 206 ² The grammar, 210 ² Removing
parentheses, 215 ² De¯ned symbols, 219 ² Pre¯x
notation (optional), 220 ² Variable sharing, 221 ² Formula schemes, 222 ² Order preserving or reversing
subformulas (postponable), 228
B Semantics 233
7 De¯nitions for semantics 235
² Interpretations, 235 ² Functional interpretations, 237
² Tautology and truth preservation, 240
8 Numerically valued interpretations 245
² The two-valued interpretation, 245 ² Fuzzy interpre-
tations, 251 ² Two integer-valued interpretations, 258
² More about comparative logic, 262 ² More about
Sugihara's interpretation, 263
9 Set-valued interpretations 269
² Powerset interpretations, 269 ² Hexagon interpretation
(optional), 272 ² The crystal interpretation, 273 ² Church's diamond (optional), 277
10 Topological semantics (postponable) 281
² Topological interpretations, 281 ² Examples, 282 ² Common tautologies, 285 ² Nonredundancy of sym-
bols, 286 ² Variable sharing, 289 ² Adequacy of ¯nite
topologies (optional), 290 ² Disjunction property (op-
tional), 293
Contents vii
11 More advanced topics in semantics 295
² Common tautologies, 295 ² Images of interpreta-
tions, 301 ² Dugundji formulas, 307
C Basic syntactics 311
12 Inference systems 313
13 Basic implication 318
² Assumptions of basic implication, 319 ² A few easy
derivations, 320 ² Lemmaless expansions, 326 ² De-
tachmental corollaries, 330 ² Iterated implication (post-
ponable), 332
14 Basic logic 336
² Further assumptions, 336 ² Basic positive logic, 339
² Basic negation, 341 ² Substitution principles, 343
D One-formula extensions 349
15 Contraction 351
² Weak contraction, 351 ² Contraction, 355
16 Expansion and positive paradox 357
² Expansion and mingle, 357 ² Positive paradox (strong
expansion), 359 ² Further consequences of positive para-
dox, 362
17 Explosion 365
18 Fusion 369
19 Not-elimination 372
² Not-elimination and contrapositives, 372 ² Interchange-
ability results, 373 ² Miscellaneous consequences of not-
elimination, 375
viii Contents
20 Relativity 377
E Soundness and major logics 381
21 Soundness 383
22 Constructive axioms: avoiding not-elimination 385
² Constructive implication, 386 ² Herbrand-Tarski De-
duction Principle, 387 ² Basic logic revisited, 393 ² Soundness, 397 ² Nonconstructive axioms and classical
logic, 399 ² Glivenko's Principle, 402
23 Relevant axioms: avoiding expansion 405
² Some syntactic results, 405 ² Relevant deduction
principle (optional), 407 ² Soundness, 408 ² Mingle:
slightly irrelevant, 411 ² Positive paradox and classical
logic, 415
24 Fuzzy axioms: avoiding contraction 417
² Axioms, 417 ² Meredith's chain proof, 419 ² Addi-
tional notations, 421 ² Wajsberg logic, 422 ² Deduction
principle for Wajsberg logic, 426
25 Classical logic 430
² Axioms, 430 ² Soundness results, 431 ² Independence
of axioms, 431
26 Abelian logic 437
F Advanced results 441
27 Harrop's principle for constructive logic 443
² Meyer's valuation, 443 ² Harrop's principle, 448 ² The disjunction property, 451 ² Admissibility, 451 ² Results in other logics, 452
Contents ix
28 Multiple worlds for implications 454
² Multiple worlds, 454 ² Implication models, 458 ² Soundness, 460 ² Canonical models, 461 ² Complete-
ness, 464
29 Completeness via maximality 466
² Maximal unproving sets, 466 ² Classical logic, 470
² Wajsberg logic, 477 ² Constructive logic, 479 ² Non-¯nitely-axiomatizable logics, 485
References 487
Symbol list 493
Index 495